Coherent Sheaves

Abstract

Let X be a scheme (or analytic space) with x ∈X and let R = O _X,x be the local ring at x. For our purposes X will usually be regular, and we could as well work in the analytic category, so that the reader can for the moment take R = ℂ{z₁,...,z _n } to be the ring of convergent power series at the origin if so desired. There are two paradigms for what a coherent shea F on X should look like:

• (i) A locally free sheaf, locally modeled on the free module R^N;

• (ii) An ideal sheaf, locally modeled on an ideal $I \subseteq R$